Required import statements and package install commands¶
In [ ]:
# pip install qiskit
# pip install qiskit_aer
# pip install qiskit-ibm-runtime

# Base
%matplotlib inline
import matplotlib.pyplot as plt
import numpy as np
from collections import Counter
import math

# Quantum
from qiskit import QuantumCircuit
from qiskit_aer import AerSimulator
from qiskit_ibm_runtime import QiskitRuntimeService
from qiskit_aer.primitives import SamplerV2 as Sampler
from qiskit.transpiler.preset_passmanagers import generate_preset_pass_manager
from qiskit.visualization import plot_histogram
Quantum circuit using Aer quantum simulator with 5 qubits and using Hadamard gates, running on CPU¶
In [ ]:
# create a smaller quantum circuit with 5 quantum bits (qubits)
qc = QuantumCircuit(5)

# apply Hadamard's gate to every qubit
qubits = [0, 1, 2, 3, 4]
qc.h(qubits)

# measure all qubits
qc.measure_all()

# use local quantum simulator (running on CPU)
aer_sim = AerSimulator()

# genertate ISA instruction set architecture
pm = generate_preset_pass_manager(backend=aer_sim, optimization_level=0)
qc_isa = pm.run(qc)

# visualize quantum circuit
qc_isa.draw('mpl')
Out[ ]:
In [ ]:
# run quantum circuit different amount of times, get results & compare them
sampler = Sampler()

shot_counts = [100, 1000, 10000, 100000, 1000000]
histograms = []

# run the circuit for each shot count and collect the results
for shots in shot_counts:
    result = sampler.run([qc], shots=shots).result()
    counts = result[0].data.meas.get_counts()
    histograms.append(counts)

# 1000 --> 1.000
# 1000000 --> 1.000.000
def formatLargeNumber(number):
    return format(number, ',').replace(',', '.')

# plot histograms in a vertical stack inside figure
fig, axes = plt.subplots(len(shot_counts), 1, figsize=(18, 16))
for i, counts in enumerate(histograms):
    plot_histogram(counts, ax=axes[i])
    axes[i].set_title(f'Quantum circuit ran {formatLargeNumber(shot_counts[i])} times', fontsize=17, fontweight='bold')
    axes[i].set_ylim(0, max(counts.values()) * 1.3)

plt.tight_layout(h_pad=3.0)
plt.show()
Quantum circuit using Aer quantum simulator with 29 qubits and using Hadamard gates, running on GPU¶
In [ ]:
# create quantum circuit with 29 quantum bits (qubits)
qc = QuantumCircuit(29)

# apply Hadamard's gate to every qubit
qubits = [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28]
qc.h(qubits)

# measure all qubits
qc.measure_all()

# use local quantum simulator (running on GPU)
aer_sim = AerSimulator(device='GPU')

# genertate ISA instruction set architecture
pm = generate_preset_pass_manager(backend=aer_sim, optimization_level=0)
qc_isa = pm.run(qc)

# visualize quantum circuit
qc_isa.draw('mpl')
Out[ ]:
In [ ]:
# run quantum circuit 1000 times & get results
sampler = Sampler()
result = sampler.run([qc_isa], shots=1000).result()

counts = result[0].data.meas.get_counts()

# print qubits and their values
for qubits, val in counts.items():
    print(f'{int(qubits, 2)}: {val}')

# plot histogram of results
plot_histogram(counts, figsize=(25,8))

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Out[ ]:
Quantum circuit using Aer quantum simulator with 29 qubits and using Rotation gates, running on GPU¶

Calculations of angles for the following probabilities: 0.25, 0.5 and 0.75


$p = 0.25$

$\sqrt{p} = \sqrt{0.25} = 0.5$

$theta = 2 * arcsin(0.5) = pi/3 = 60°$


$p = 0.5$

$\sqrt{p} = \sqrt{0.5} = 0.707$

$theta = 2 * arcsin(0.707) = pi/2 = 90°$


$p = 0.75$

$\sqrt{p} = \sqrt{0.75} = 0.866$

$theta = 2 * arcsin(0.866) = 2pi/3 = 120°$

In [ ]:
# create quantum circuit with 29 qubits
qc = QuantumCircuit(29)

qubits = [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28]

# define probabilities
probabilities = [0.25, 0.5, 0.75]

# calculate rotation angles in radians and convert to degrees
angles_rad = [round(2 * math.asin(math.sqrt(p)), 2) for p in probabilities]
angles_deg = [round(math.degrees(angle)) for angle in angles_rad]

print(f'angles in radians: {angles_rad}')
print(f'angles in degrees: {angles_deg}')

# pi/3 (60°) = 25 %
# pi/2 (90°) = 50 %
# 2*pi/3 (120°) = 75 %

# rotation gates set to pi/2 (50%) probability for each qubit
qc.rx(math.pi/2, qubits)

# measure all qubits
qc.measure_all()

# use local quantum simulator (running on GPU)
aer_sim = AerSimulator(device='GPU')

# genertate ISA instruction set architecture
pm = generate_preset_pass_manager(backend=aer_sim, optimization_level=0)
qc_isa = pm.run(qc)

# visualize quantum circuit
qc_isa.draw('mpl')

angles in radians: [1.05, 1.57, 2.09]
angles in degrees: [60, 90, 120]
Out[ ]:
In [ ]:
# run quantum circuit 1000 times & get results
sampler = Sampler()
result = sampler.run([qc_isa], shots=1000).result()

counts = result[0].data.meas.get_counts()

# print qubits and their values
for qubits, val in counts.items():
    print(f'{int(qubits, 2)}: {val}')

# plot histogram of results
plot_histogram(counts, figsize=(25,8))

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Out[ ]:

Calculation of angle (in radians) for generating number 3 with a 50 % probability¶

3(10) = 11(2)

$P(11) = p * p = p^2$

$p^2 = 0.5$

$p = \sqrt{0.5} = 1/\sqrt{2} = 0.707$

$p = sin^2(\theta/2)$

$sin^2(\theta/2) = 1/\sqrt(2)$

$sin(\theta/2) = \sqrt{1/\sqrt{2}} = 0.841$

$\theta/2 = arcsin(0.841) = 57° = 1.0$ radian

$\theta = 2 * 57 = 114.5° = 2.0$ radian

In [ ]:
# create quantum circuit with 2 qubits (number 3 in decimal only requires 2 binary digits)
qc = QuantumCircuit(2)

qubits = [0, 1]
theta = 2

# rotation gates set to 2 radians probability for each of the 2 qubits
qc.rx(theta, qubits)

# measure all qubits
qc.measure_all()

# use local quantum simulator (running on GPU)
aer_sim = AerSimulator(device='GPU')

# genertate ISA instruction set architecture
pm = generate_preset_pass_manager(backend=aer_sim, optimization_level=0)
qc_isa = pm.run(qc)

# visualize quantum circuit
qc_isa.draw('mpl')
Out[ ]:
In [ ]:
# run quantum circuit 1000 times & get results
sampler = Sampler()
result = sampler.run([qc_isa], shots=100).result()

counts = result[0].data.meas.get_counts()

for bits, val in counts.items():
    print(f'{int(bits, 2)}: {val}')

# plot histogram of results
plot_histogram(counts)

0: 11
3: 46
1: 22
2: 21
Out[ ]:

Calculation of angle (in radians) for generating 576(10) = 1001000000(2) with a 25 % probability¶

Since only 2 bits in the binary representation of the decimal number 576 are set to 1, we can do the following:

$P = 0.25$

$P = p^2$ (power of 2 because only 2 bits are set to 1)

$p^2 = 0.25$

$p = \sqrt{0.25} = 0.5$

$p = sin^2(\theta/2)$

$sin^2(\theta/2) = 0.5$

$sin(\theta/2) = \sqrt{0.5} = 0.707$

$\theta/2 = arcsin(0.707) = 45° = 0.785$ radian

$\theta = 2 * 45 = 90° = 1.57$ radian

In [ ]:
# create quantum circuit with 10 qubits (number 576 in decimal requires 10 binary digits)
qc = QuantumCircuit(10)

# 576 --> 1001000000
# only 6th and 9th bit are set to 1 (if counting from the back to the front)
qubits = [6, 9] 
# calculated theta in radians
theta = 1.57

# rotation gates set to 2 radians probability for each of the 2 qubits
qc.rx(theta, qubits)

# measure all qubits
qc.measure_all()

# use local quantum simulator (running on GPU)
aer_sim = AerSimulator(device='GPU')

# genertate ISA instruction set architecture
pm = generate_preset_pass_manager(backend=aer_sim, optimization_level=0)
qc_isa = pm.run(qc)

# visualize quantum circuit
qc_isa.draw('mpl')
Out[ ]:
In [ ]:
# run quantum circuit 1000 times & get results
sampler = Sampler()
result = sampler.run([qc_isa], shots=1000).result()

counts = result[0].data.meas.get_counts()

for bits, val in counts.items():
    print(f'{int(bits, 2)}: {val}')

# plot histogram of results
plot_histogram(counts)

512: 245
0: 228
576: 289
64: 238
Out[ ]:

Calculation of angle (in radians) for generating 576(10) = 1001000000(2) with a 50 % probability¶

Since only 2 bits in the binary representation of the decimal number 576 are set to 1, we can do the following:

$P = 0.5$

$P = p^2$ (power of 2 because only 2 bits are set to 1)

$p^2 = 0.5$

$p = \sqrt{0.5} = 0.707$

$p = sin^2(\theta/2)$

$sin^2(\theta/2) = 0.707$

$sin(\theta/2) = \sqrt{0.707} = 0.841$

$\theta/2 = arcsin(0.841) = 57° = 1.0$ radian

$\theta = 2 * 57 = 114.5° = 2.0$ radian

In [ ]:
# create quantum circuit with 10 qubits (number 576 in decimal requires 10 binary digits)
qc = QuantumCircuit(10)

# 576 --> 1001000000
# only 6th and 9th bit are set to 1 (if counting from the back to the front)
qubits = [6, 9]
# calculated theta in radians
theta = 2

# rotation gates set to 2 radians probability for each of the 2 qubits
qc.rx(theta, qubits)

# measure all qubits
qc.measure_all()

# use local quantum simulator (running on GPU)
aer_sim = AerSimulator(device='GPU')

# genertate ISA instruction set architecture
pm = generate_preset_pass_manager(backend=aer_sim, optimization_level=0)
qc_isa = pm.run(qc)

# visualize quantum circuit
qc_isa.draw('mpl')
Out[ ]:
In [ ]:
# run quantum circuit 1000 times & get results
sampler = Sampler()
result = sampler.run([qc_isa], shots=1000).result()

counts = result[0].data.meas.get_counts()

for bits, val in counts.items():
    print(f'{int(bits, 2)}: {val}')

# plot histogram of results
plot_histogram(counts)

64: 206
576: 498
512: 199
0: 97
Out[ ]:

Calculation of angle (in radians) for generating 576(10) = 1001000000(2) with a 75 % probability¶

Since only 2 bits in the binary representation of the decimal number 576 are set to 1, we can do the following:

$P = 0.75$

$P = p^2$ (power of 2 because only 2 bits are set to 1)

$p^2 = 0.75$

$p = \sqrt{0.75} = 0.866$

$p = sin^2(\theta/2)$

$sin^2(\theta/2) = 0.866$

$sin(\theta/2) = \sqrt{0.866} = 0.931$

$\theta/2 = arcsin(0.931) = 68.6° = 1.2$ radian

$\theta = 2 * 68.6 = 137.2° = 2.4$ radian

In [ ]:
# create quantum circuit with 10 qubits (number 576 in decimal requires 10 binary digits)
qc = QuantumCircuit(10)

# 576 --> 1001000000
# only 6th and 9th bit are set to 1 (if counting from the back to the front)
qubits = [6, 9]
# calculated theta in radians
theta = 2.4

# rotation gates set to 2 radians probability for each of the 2 qubits
qc.rx(theta, qubits)

# measure all qubits
qc.measure_all()

# use local quantum simulator (running on GPU)
aer_sim = AerSimulator(device='GPU')

# genertate ISA instruction set architecture
pm = generate_preset_pass_manager(backend=aer_sim, optimization_level=0)
qc_isa = pm.run(qc)

# visualize quantum circuit
qc_isa.draw('mpl')
Out[ ]:
In [ ]:
# run quantum circuit 1000 times & get results
sampler = Sampler()
result = sampler.run([qc_isa], shots=1000).result()

counts = result[0].data.meas.get_counts()

for bits, val in counts.items():
    print(f'{int(bits, 2)}: {val}')

# plot histogram of results
plot_histogram(counts)

64: 108
576: 751
512: 119
0: 22
Out[ ]:

Kaj ste morali spremeniti v algoritmu, da ste dosegli poljubno verjetnost generiranja naključnih števil?¶

Z uporabo enačb sem moral za poljubno verjetnost izračunati kot theta (v radianih).

Ali bi lahko zgoraj navedene poljubne verjetnosti (0,25, 0,50 in 0,75), dosegli s pomočjo Hadamardovih vrat? Odgovor utemeljite.¶

Z uporabo Hadamardovih vrat smo pri generaciji s poljubnimi verjetnostmi omejeni, in sicer lahko s Hadamardovimi vrati generiramo samo s 50-odstotno verjetnostjo. Razlog za to je narava njihovega delovanja, saj Hadamardova vrata generirajo samo ekvivalentno superpozicijo qubitov $|0\rangle$ in $|1\rangle$.

S kakšno verjetnostjo se v vašem generatorju generirajo ostala števila (poleg tistega, ki ga določajo zadnje tri cifre vaše vpisne številke)?¶

Števila, ki imajo v binarni obliki na enakih mestih (kot število 576) bite nastavljene na 1, so generirana z večjo verjetnostjo, kot tista, ki se s številom 576 v bitih ne ujemajo.

Large quantum circuit using IBM's quantum simulator with 127 qubits¶
In [ ]:
# create a larger quantum circuit with 127 quantum bits (qubits)
qc = QuantumCircuit(127)

# apply Hadamard's gate to every qubit
qubits = [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32,
          33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64,
          65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96,
          97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126]

qc.h(qubits)

# measure all qubits
qc.measure_all()

# use IBM's quantum computer simulator (register here to get API TOKEN: https://quantum.ibm.com/)
API_TOKEN = "1ac02f42d976d14a0cea984659aa1d1d13bbf4746537030ad3ee4912f7ac00f4d3fa678570de8ce909d2b30ed887892f03e92f05c306ac94229dce2dbfe2f830"
provider = QiskitRuntimeService(channel='ibm_quantum', token=API_TOKEN)
backend = provider.backend("ibm_kyoto")

# genertate ISA instruction set architecture
pm = generate_preset_pass_manager(backend=backend, optimization_level=0)
qc_isa = pm.run(qc)

# visualize quantum circuit
qc_isa.draw('mpl')
Out[ ]:
In [ ]:
# run quantum circuit 1000 times & get results
sampler = Sampler()
result = sampler.run([qc_isa], shots=1000).result()

counts = result[0].data.meas.get_counts()

# print qubits and their values
for qubits, val in counts.items():
    print(f'{int(qubits, 2)}: {val}')

# plot histogram of results
plot_histogram(counts, figsize=(30,8))

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